Solid propellants — AP - HTPB - Al composition; burn rate dependence on pressure (Vieille's law)

5.3.8 · Chemistry › Combustion Chemistry (Propulsion Bridge)

1. What is a composite solid propellant?

WHY each component?

WHY solid mein oxidiser? Rocket ke andar koi hawa nahi hoti. Propellant ko apna khud ka oxygen saath laana padta hai → AP decompose hokar $O_{2}$ , $C l$ , $H_{2} O$ , $N_{2}$ release karta hai.
WHY HTPB (ek rubber)? Yeh dono kaam karta hai — fuel bhi hai (C, H jalte hain) aur glue bhi jo ek castable, flexible grain banata hai jo acceleration mein crack nahi karti.
WHY aluminium? $A l + \frac{3}{4} O_{2} \to \frac{1}{2} A l_{2} O_{3}$ enormously exothermic hai. Yeh flame temperature (~3500 K) badhata hai aur exhaust mein mass add karta hai, thrust increase karta hai. $A l_{2} O_{3}$ ka smoke hi wajah hai ki aise motors ek thick white trail chhodte hain.

2. Burn rate aur burning surface

WHY pressure matter karta hai? Zyada chamber pressure $P$ → flame zone surface ke karib push hoti hai aur reaction rates badhte hain → surface par zyada heat flux wapas → faster regression. Toh $r$ $P$ ke saath badhta hai.

3. Vieille's Law (Saint Robert's law) — derived

First principles se derivation (heat-flux balance):

Solid tab consume hota hai jab flame itni heat deliver kare ki surface $T_{0}$ se ignition $T_{s}$ tak raise ho jaye. Energy balance per unit area: $ρ_{p} r c (T_{s} - T_{0}) = q^{''} (conductive flux from flame)$ Yeh step kyun? Area per mass flux $ρ_{p} r$ hai; har kg ko heat up hone ke liye $c (T_{s} - T_{0})$ joules chahiye.
Conductive flux hai $q^{''} = \frac{λ ( T _{f} - T _{s} )}{δ}$ , jahan $δ$ flame stand-off distance hai. Kyun? Thickness $δ$ ke gas gap mein Fourier's law.
Flame thickness reaction rate ke saath scale karti hai. Pressure se related overall reaction order ke liye, reaction rate per volume $\propto P^{m}$ , aur gas density $\propto P$ , toh $δ \propto P^{- k}$ (flame high $P$ par karib aati hai). Kyun? Faster chemistry + denser gas = thinner, closer flame.
Substitute karo: $ρ_{p} r c (T_{s} - T_{0}) = \frac{λ ( T _{f} - T _{s} )}{δ} \propto P^{k}$ $\Rightarrow r \propto P^{k} \equiv a P^{n}$ jahan $n = k$ aur $a$ thermal/chemical constants ko bundle karta hai (jo initial temperature $T_{0}$ par depend karte hain).

4. Equilibrium chamber pressure — kyun $n < 1$ mandatory hai

Figure — Solid propellants — AP - HTPB - Al composition; burn rate dependence on pressure (Vieille's law)

5. Worked examples

Recall Feynman: 12 saal ke bachche ko explain karo

Ek solid rocket ek giant firework stick hai. Hum "woh cheez jo jalti hai" (rubber + shiny aluminium dust) ko "woh cheez jo oxygen deti hai" (ek white salt) ke saath mix karke ek hard rubbery candle mein bake karte hain. Jab hum use jalate hain, surface inch by inch jal ke khatam hoti hai. Andar ke gas par zyada pressure daalo (zyada pressure) toh yeh tezi se jalti hai — lekin agar har baar pressure badhne par yeh bahut zyada tezi se jalti hai, toh BANG ho jaata hai. Toh engineers ek aisa recipe chunte hain jahan speeding up gentle aur controllable ho. Rule "burn speed = $a$ times pressure-to-the-power- $n$ " bas woh dial hai jo batata hai ki candle khud ko kitni tezi se kha rahi hai.

Active Recall Flashcards

AP ka matlab kya hai aur composite propellant mein iska role kya hai?

Ammonium perchlorate

N H_{4} C l O_{4}

; yeh oxidiser hai, combustion ke liye oxygen supply karta hai.

HTPB kya karta hai?

Hydroxyl-terminated polybutadiene — rubbery fuel-binder jo grain ko saath hold karta hai AUR fuel ki tarah bhi jalta hai.

AP/HTPB propellants mein aluminium kyun add kiya jaata hai?

Yeh ek high-energy metal fuel hai;

A l \to A l_{2} O_{3}

strongly exothermic hai, flame temperature aur specific impulse badhata hai.

Vieille's (Saint Robert's) law batao.

r = a P^{n}

— burn rate ek constant times chamber pressure to the power n ke barabar hai.

Exponent

n

physically kya represent karta hai?

Pressure ke liye burn rate ki sensitivity.

Stable motor ke liye

n < 1

kyun zaroori hai?

Equilibrium pressure

\propto (A_{b} / A_{t})^{1/ (1 - n)}

; agar

n \geq 1

toh exponent diverge karta hai → runaway pressure → explosion.

Do (P, r) data points se

n

kaise nikaalte hain?

n = ln (r_{2} / r_{1}) / ln (P_{2} / P_{1})

— ratio

a

ko cancel karta hai, logs slope dete hain.

Burning grain ki mass generation rate likho.

\overset{m}{˙} = ρ_{p} A_{b} r

Zyada pressure burn rate kyun badhata hai (physical mechanism)?

Zyada P flame ko surface ke karib push karta hai aur chemistry speed up karta hai → zyada heat flux conduct karke wapas aata hai → faster surface regression.

AP/HTPB propellants ke liye typical n range kya hai?

Lagbhag 0.2 se 0.5.

Connections

Vieille's Law — burn-rate power law detail mein
Specific Impulse — Al $I_{s p}$ kyun badhata hai
Characteristic Velocity c-star — equilibrium pressure derivation mein aata hai
Combustion Thermodynamics — flame temperature & adiabatic flame T
Liquid Propellants — contrast: throttleable vs non-throttleable
Heat Conduction (Fourier's Law) — burn-rate derivation ka basis
Ammonium Perchlorate Decomposition — oxidiser chemistry